[GAP Forum] Basic question, order of fpgroup

[K C H Mackenzie]

Dear Stefan,

Thanks very much. That's very helpful.

That it's infinite is very surprising. I believed G to be an extension of
S_5
by a finite product of cyclic groups of order 2. I need to think a lot more
about
this.

Best, Kirill

On 27 July 2011 17:22, Stefan Kohl <stefan at mcs.st-and.ac.uk> wrote:

> Dear Forum,
>
> Kirill Mackenzie asked:
>
> [ ... ]
>
> > f:=FreeGroup(4);;
> > G:=f/[f.1^2,f.2^2,f.3^2,f.4^2,
> > (f.1*f.2)^3,(f.1*f.3)^3,(f.1*f.4)^3,(f.2*f.3)^3,(f.2*f.4)^3,(f.3*f.4)^3,
> > (f.1*f.2*f.1*f.3)^4,(f.1*f.2*f.1*f.4)^4,
> > (f.1*f.3*f.1*f.4)^4,(f.1*f.3*f.1*f.2)^4,
> > (f.1*f.4*f.1*f.2)^4,(f.1*f.4*f.1*f.3)^4,
> > (f.2*f.3*f.2*f.4)^4,(f.2*f.3*f.2*f.1)^4,
> > (f.2*f.4*f.2*f.1)^4,(f.2*f.4*f.2*f.3)^4,
> > (f.2*f.1*f.2*f.3)^4,(f.2*f.1*f.2*f.4)^4,
> > (f.3*f.4*f.3*f.2)^4,(f.3*f.4*f.3*f.1)^4,
> > (f.3*f.1*f.3*f.2)^4,(f.3*f.1*f.3*f.4)^4,
> > (f.3*f.2*f.3*f.1)^4,(f.3*f.2*f.3*f.4)^4,
> > (f.4*f.1*f.4*f.2)^4,(f.4*f.1*f.4*f.3)^4,
> > (f.4*f.2*f.4*f.1)^4,(f.4*f.2*f.4*f.3)^4,
> > (f.4*f.3*f.4*f.2)^4,(f.4*f.3*f.4*f.1)^4];
>
> > To start I want the order of G. With `Size(G);' `Order(G);'
> `IsFinite(G);'
> > I just get several (8 or 9) iterations of
> >
> > #I  Coset table calculation failed -- trying with bigger table limit
> >
> > and then
> >
> > exceeded the permitted memory (`-o' command line option) at
>
> A common strategy to decide finiteness of an fp group which is
> 'often' successful is to search for low-index subgroups which
> have the infinite cyclic group as an homomorphic image, i.e.
> which have 0 among their abelian invariants:
>
> gap> low := LowIndexSubgroupsFpGroup(G,20);;
> gap> Set(List(low,AbelianInvariants));
> [ [  ], [ 0, 0, 0, 2 ], [ 0, 0, 2, 2 ], [ 2 ], [ 2, 2 ], [ 2, 2, 2, 2 ],
>  [ 2, 2, 2, 2, 2 ], [ 2, 2, 2, 2, 3 ], [ 2, 2, 2, 2, 4 ], [ 2, 2, 2, 4 ],
>  [ 2, 2, 4 ], [ 2, 2, 8 ], [ 2, 4, 4, 4 ], [ 3 ] ]
>
> >From this you see that your group G has such subgroups,
> thus it is infinite.
>
> Further you can compute finite quotients of your group
> by letting it act by multiplication from the right on the
> right cosets of a subgroup. -- For example:
>
> gap> quots := List(low,H->Action(G,RightCosets(G,H),OnRight));;
> gap> List(quots,Size);
> [ 1, 120, 1920, 1920, 2432902008176640000, 1920, 1920, 1920,
>  2432902008176640000, 1920, 1920, 2432902008176640000, 1920, 1920, 1920,
>  1920, 1920, 1920, 2432902008176640000, 1920, 1920, 2432902008176640000,
>  1920, 1920, 1920, 1920, 120, 3840, 3840, 3840, 3840, 3840, 3840, 3840,
>  1920, 1920, 1920, 1920, 1920, 3840, 3840, 2432902008176640000, 3840, 3840,
>  3840, 3840, 3840, 120, 1920, 1920, 1920, 1920, 120, 3840, 3840, 3840,
> 3840,
>  3840, 3840, 3840, 120, 2, 120, 120, 3840, 3840, 3840, 3840, 3840, 3840,
>  3840, 120, 3840, 95040, 3840, 3840, 239500800, 3840, 3840, 95040, 3840,
>  3840, 3840, 3840, 95040, 3840, 239500800, 239500800, 3840, 95040, 3840,
>  95040, 3840, 239500800, 3840, 95040, 120 ]
>
> For example you see that your group has a quotient which is
> isomorphic to the Mathieu group M12:
>
> gap> List(Filtered(quots,IsSimple),StructureDescription);
> [ "C2", "M12", "A12", "M12", "M12", "A12", "A12", "M12", "M12", "A12",
> "M12" ]
>
> Hope this helps,
>
>    Stefan Kohl
>
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